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### Concept of synchronizability

This thread is about the concept of synchronizability. It's a concept I tried to formalize in its most general sense but without success. The goal of this thread is therefore to try to formalize it in ...

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### Idempotent solutions to the implict function theorem other than the identity?

I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...

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### Which limit to take as a key applied math decision

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...

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### Seminar Notes Repository

On Seminars, people actually talk a lot more about motivations than when they write in paper. It would be a good idea if there is an online repository where people can upload notes (handwritten, ...

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### Reference request: name of a transform

Define a transform on polynomials which is linear and acts on each monomial as $$\widehat{z^k} = \frac{(1+z)(2+z)\ldots(k+z)}{k!}.$$ Does anyone know whether this has a name (and therefore has been ...

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### Find Moment condition for generalized method of moments

Consider a scalar system with 2K outputs and K+2 unknowns
$y_{k,1}=x_ka_1+n_{k,1} \quad y_{k,2}=x_ka_2+n_{k,1}$.
The variables $n_{k,\ell}$ are zero mean noise variables.
To estimate $a_1$ and $a_2$, ...

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### Jacobi triple product for multidimensional lattices

The Jacobi triple product identity gives as a special case a product formula for the theta function of a 1-dimensional lattice. Is there a more general product formula for the theta function of an ...

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### Finite topological dimension x local compactness

Of course, the two notions are independent one from the other, but often one of them implies the other under some additional hypotheses. For instance:
A topological vector space is finite dimensional ...

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### General procedure to find the determinant of an operator?

I want to learn to find the determinant of an operator.
I am given an operator like
$\Sigma _{\alpha\beta}=-k^2g_{\alpha\beta}+i\theta\epsilon_{\alpha\beta\gamma} k^\gamma$
$k^2=k^μk_μ$, $g^{αβ}$ ...

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### Conjectures on fractions where each digit appears once in numerator and denominator

This is a highly redacted version of a question that was asked before. Please see Criteria of considering relevance of the question to the domain of research topics for details.
Some numerical ...

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### Reference needed for: Automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning

I am doing research on automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning.
I have found several research papers on ...

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### Allocation game optimal strategy

There are two players, Alice and Bob. There is an initial pool of 100 dollars. Alice proposes an allocation of the dollars (real numbers, not necessarily integers), and Bob can either accept or ...

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### Regarding a Feature of Multivariate Real Function

Any real function can be expressed as a function of the sum of two monotonic real functions?
More precisely, for real function p(x, y), there exist continuous real functions P(x), h(x,y), g(x) such ...

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### Asymptotic inverses of asymptotic functions

The prime number theorem states that two functions are asymptotic. Their inverses (as functions of an integral variable) are also asymptotic. In general, under what conditions are the inverses of ...

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### Repeated function resulting in quadratic time.

Let $$r(f, x) = k$$ such that
$$f^k (x) < 2 \hbox{ and } f^{k-1} (x) \geq 2$$
For example $$r(n \rightarrow n-1, 2^n) = 2^n-1; r(n \rightarrow n/2, 2^n) = n.$$
For an arbitrary $c$, we have $$r( ...